Thoma type results for discrete quantum groups

Abstract

Thoma's theorem states that a group algebra [Formula: see text] is of type I if and only if [Formula: see text] is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group [Formula: see text], we have a stationary model of type [Formula: see text], with [Formula: see text] being a finite quantum group, and with [Formula: see text] being a compact group. We discuss then some refinements of these results in the quantum permutation group case, [Formula: see text], by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank [Formula: see text].